DOCSLIB.ORG

A Card Game for Bell's Theorem and Its Loopholes Guy Vandegrift¹, Joshua Stomel²

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Card Game for Bell's Theorem and Its Loopholes Guy Vandegrift¹, Joshua Stomel² WikiJournal of Science, 2018, 1(1):6 doi: 10.15347/wjs/2018.005 Research Article A card game for Bell's theorem and its loopholes Guy Vandegrift¹, Joshua Stomel² Abstract In 1964 John Stewart Bell made an observation about the behavior of particles separated by macroscopic distances that had puzzled physicists for at least 29 years, when Einstein, Podolsky and Rosen put forth the famous EPR paradox. Bell made certain assumptions leading to an inequality that entangled particles are routinely observed to violate in what are now called Bell test experiments. As an alternative to showing students a "proof" of Bell's inequality, we introduce a card game that is impossible to win. The solitaire version is so simple it can be used to introduce binomial statistics without mentioning physics or Bell's theorem. Things get interesting in the partners' version of the game because Alice and Bob can win, but only if they cheat. We have identified three cheats, and each corresponds to a Bell's theorem "loophole". This gives the instructor an excuse to discuss detector error, cau- sality, and why there is a maximum speed at which information can travel. The conundrum any theory that might replace quantum mechanics.[6] It is also possible to view a loophole as a physical mecha- Although this can be called a theorem, it might be bet- nism by which the outcome of a Bell's theorem experi- ter viewed as something "spooky" that has been rou- ment might seem less "spooky". In this paper, we asso- tinely observed, and is consistent with quantum me- ciate loopholes with ways to cheat at the partners' ver- chanics. But this puzzling behaviour violates what sion of the card game. It should be noted that the might be called common notions about what is and is three loophole mechanisms introduced in this paper not possible.[1][2] Students typically encounter a math- raise questions that are even spookier than quantum ematical theorem as an incomprehensible statement mechanics: Are the photons "communicating" with each that cannot be digested until it is first proven and then other? Do they "know" the future? Do they "persuade" applied in practice. It is not uncommon for novices to the measuring devices to fail when the "cards are unfa- refer to some version of Bell's inequality as Bell's theo- vorable"?[7] rem because the inequality can be mathematically "proven".[3] The problem is that what is proven turns out Since entanglement is so successfully modeled by to be untrue. quantum mechanics, one can argue that there is no need for a mechanism that "explains" it. Nevertheless, David Mermin described an imaginary device not unlike there are reasons for investigating loopholes. At the that shown in Fig. 1, and refers to the fact that such a most fundamental level, history shows that a successful device actually exists as a conundrum, then pointed out physical theory can be later shown to be an approxima- that many physicists deny that it is a conundrum.[4] tion to a deeper theory, and the need for this new the- ory is typically associated with a failure of the old para- A simple Bell's theorem experiment digm. It is plausible that a breakdown of quantum me- It is customary to name the particles[5] in a Bell's theo- chanics might be discovered using a Bell's theorem ex- rem experiment "Alice" and "Bob", an anthropomor- periment designed to investigate a loophole. But the phism that serves to emphasize the fact that a pair of vast majority of us (including most working physicists) humans cannot win the card game ... unless they need other reasons to care about loopholes: Many find cheat. To some experts, a "loophole" is a constraint on it interesting that we seem to live in a universe gov- erned by fundamental laws, and Bell's theorem yields insights into the bizarre nature of those laws. Also, 1 Wright State University Lake Campus, Celina, Ohio, 45822 those who teach can use these card games to motivate 2 Wright State University Lake Campus, Celina, Ohio, 45822 *Author correspondence: [email protected] ORCID: [0000-0000-0000-0001] Supplementary material: commons.wikimedia.org/location Licensed under: CC-BY Received 02-02-2018; accepted 31-05-2018 1 of 10 | WikiJournal of Science WikiJournal of Science, 2018, 1(1):6 doi: 10.15347/wjs/2018.005 Research Article introductory discussions about statistical inference, po- while the odd cards (3,5,7,9) represent a photon being larization, and modern physics. blocked. Although Bell's inequality is easy to prove[9], we avoid it here because the card game reverses roles regarding probability: Instead of the investigators attempting to ascertain the photons' so-called hidden variables, the players are acting as particles attempting to win the game by guessing the measurement angles. Another Figure 1 | The outside casing of each device remains station- complication is that the original form of Bell's inequal- ary while the circle with parallel lines rotates with the center ity does not adequately model the partners' version of arrow pointing in one of three directions (♥, ♣, ♠.) If Jacks the game because humans have the freedom to exhibit are used to represent these directions, Alice will see J♥ as her question card. She will respond with an "odd"-numbered a behavior not observed by entangled particles (under answer card (3♥) to indicate that she is blocked by the filter. ideal experimental conditions). This behavior involves a If Bob passes through a filter with the "spade" orientation, 100% correlation (or anti-correlation) whenever the po- he sees J♠ as the question card, and answers with the "even" larization measurement angles are either parallel or numbered 2♠. This wins one point for the team because perpendicular to each other.[10] In the partners' version they gave different answers to different questions. of the card game, this behavior must be enforced by de- ducting a penalty from the partners' score whenever Figure 1 shows a hypothetical and idealized experi- they are caught using a forbidden strategy (which we ment involving two entangled photons simultaneously shall later call the β-strategy). The minimum required emitted by a single (parent) atom. After the photons penalty is calculated in Supplementary file:The car have been separated by some distance, each is ex- and the goats. Fortunately students need not master posed to a measurement that determines whether the this calculation because the actual penalty should often photon would pass or be blocked by the polarizing fil- be whatever it takes to encourage a strategy that mim- ter.[8] To ensure that the results seem "spooky" it ics this aspect of entanglement (which we shall call the should be possible to rotate the filter while the pho- α-strategy.) tons are en route so that the filter's angle of orienta- A theoretical understanding of how one can model en- tion is not "known" to either photon until it encounters tanglement using the Schrödinger equation can be the filter. If the filters are rotated between only three found in Supplementary file:Tube entanglement. polarization angles, we may use card suits (hearts ♥, clubs ♣, spades ♠) to represent these angles. These The solitaire card game three polarization angles are associated with "ques- Figure 2 shows the three possible outcomes associated tion" cards, because the measurement essentially asks with one hand of the solitaire version of the game. The the photon a question: solitaire version requires nine cards. The figure uses a set with three "jacks" (♥ ♣ ♠) for the questions, and (2,3) "Will you pass through a filter oriented at for the six (even/odd) answer cards. To play one round this angle?" of the game, the player first shuffles the three question For simplicity we restrict our discussion to symmetric cards and places them face down so their identity is not angles (0°, 120°, 240°.) The filter's axis of polarization known. Next, for each of the three suits, the player se- is shown in the figure as parallel lines, with the center lects an even or odd answer card. The figure shows the line pointing to the heart, club, or spade. Any face card player choosing the heart and club to be even, while the can be used to "ask" the question, and the four face spade is odd: 2♥ 2♣ 3♠. This is the only viable strategy, cards (jack, queen, king, ace) are equivalent. If the de- since the alternative is to always lose by selecting three tectors are flawless, each measurement is binary: The answers that are all even or all odd. In the partner's ver- photon either passes or is blocked by the filter (subse- sion we shall introduce a second, β-strategy, which is quent measurements on a photon would yield nothing not possible in the solitaire game. interesting.) The measurement's outcome is repre- After three answer cards are selected and turned face sented by an even or odd numbered "answer" card (of up, two of the three question cards are randomly se- the same suit). The numerical value of an answer card lected and also turned face up. Figure 2 depicts all three is not important: all even numbers (2,4,6,8) are equiva- equally probable outcomes, or ways to select two out of lent and represent a photon passing through the filter, 2 of 10 | WikiJournal of Science WikiJournal of Science, 2018, 1(1):6 doi: 10.15347/wjs/2018.005 Research Article three cards (3 choose 2.)[11] The round is scored by add- with a probability of 3/4. Table 1 shows that this scoring ing or subtracting points, as shown in Table 1: First the system causes humans to average a loss of at least 1/3 suit of each of the two upturned question cards is of a point per round, while entangled particles maintain matched to the corresponding answer card.
Load more

Recommended publications
  • A Question of Quantum Reality 24 September 2020
    A question of quantum reality 24 September 2020 virtuality. Bell spent most of his working life at CERN in Geneva, Switzerland, and Bertlmann first met him when he took up a short-term fellowship there in 1978. Bell had first presented his theorem in a seminal paper published in 1964, but this was largely neglected until the 1980s and the introduction of quantum information. Bertlmann discusses the concept of Bell inequalities, which arise through thought experiments in which a pair of spin-½ particles propagate in opposite directions and are measured Credit: Pixabay/CC0 Public Domain by independent observers, Alice and Bob. The Bell inequality distinguishes between local realism—the 'common sense' view in which Alice's observations do not depend on Bob's, and vice versa—and Physicist Reinhold Bertlmann of the University of quantum mechanics, or, specifically, quantum Vienna, Austria has published a review of the work entanglement. Two quantum particles, such as of his late long-term collaborator John Stewart Bell those in the Alice-Bob situation, are entangled of CERN, Geneva in EPJ H. This review, "Real or when the state measured by one observer Not Real: that is the question," explores Bell's instantaneously influences that of the other. This inequalities and his concepts of reality and theory is the basis of quantum information. explains their relevance to quantum information and its applications. And quantum information is no longer just an abstruse theory. It is finding applications in fields as John Stewart Bell's eponymous theorem and diverse as security protocols, cryptography and inequalities set out, mathematically, the contrast quantum computing.
    [Show full text]
  • Final Copy 2020 11 26 Stylia
    This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Stylianou, Nicos Title: On 'Probability' A Case of Down to Earth Humean Propensities General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. On ‘Probability’ A Case of Down to Earth Humean Propensities By NICOS STYLIANOU Department of Philosophy UNIVERSITY OF BRISTOL A dissertation submitted to the University of Bristol in ac- cordance with the requirements of the degree of DOCTOR OF PHILOSOPHY in the Faculty of Arts.
    [Show full text]
  • Disproof of John Stewart Bell
    Disproof of John Stewart Bell T. Liffert 4th October 2020 Abstract The proof of John Stewart Bell [1] [2] is based on wrong premises. A deterministic model for a hidden mechanism with an parameter λ is therefor possible. This model is possible by an integration of the experimental context into the wave function 1 The basic experiment The following considerations are based on an experiment with a pair of en- tangled photons, each of them meets a polarisation filter. The crucial phe- nomenon consists of the fact, that, if the polarisation filters are adjusted identically, the photons react always identically: Either both will be ab- sorbed or both will be transmitted. For the angular difference Φ between the polarisation filter adjustments the probability, that both photons give identi- cal measurement results, is the square of the cosine of that angular difference (law of Malus)[3]. The set of the possible measurement results is a binary one, it can be referred to by expressions like f0; 1g or f+1; −1g. For later considerations of serie experiments and related expected values I use the set f+1; −1g, for the treatment of the proof variant according to Wigner-Bell I choose the set f0; 1g. I use the following assignments for measurement results and eigenstates • 0 j0i ! 1 • 1 j1i ! −1 May α denote the measurement result of the left photon and β the one of the right photon. The upper proposition for the conditional probability P of identical measurement results given the angular difference Φ one would write like this: P (α = βjΦ) = cos2 Φ (1) 1 1.1 The quantum mechanical doctrine In the literature that I know there is a strict separation between the quantum state jΨi of the pair of photons on the one hand and the experimental context, consisting of nothing more than the angular difference of the polarisation filters or the polarizers, on the other hand.
    [Show full text]
  • John S. Bell's Concept of Local Causality
    John S. Bell’s concept of local causality Travis Norsena) Department of Physics, Smith College, McConnell Hall, Northampton, Massachusetts 01063 (Received 15 August 2008; accepted 6 August 2011) John Stewart Bell’s famous theorem is widely regarded as one of the most important developments in the foundations of physics. Yet even as we approach the 50th anniversary of Bell’s discovery, its meaning and implications remain controversial. Many workers assert that Bell’s theorem refutes the possibility suggested by Einstein, Podolsky, and Rosen (EPR) of supplementing ordinary quantum theory with “hidden” variables that might restore determinism and/or some notion of an observer- independent reality. But Bell himself interpreted the theorem very differently—as establishing an “essential conflict” between the well-tested empirical predictions of quantum theory and relativistic local causality. Our goal is to make Bell’s own views more widely known and to explain Bell’s little- known formulation of the concept of relativistic local causality on which his theorem rests. We also show precisely how Bell’s formulation of local causality can be used to derive an empirically testable Bell-type inequality and to recapitulate the EPR argument. VC 2011 American Association of Physics Teachers. [DOI: 10.1119/1.3630940] I. INTRODUCTION theory could be rendered compatible with local causality by following Newton and by denying that the theory in question In its most general sense, “local causality” is the idea that provided a complete description of the relevant phenomena. physical influences propagate continuously through space— This changed in 1905 with Einstein’s discovery of special that what Einstein famously called “spooky actions at a dis- relativity, which for the first time identified a class of causal 1 tance” are impossible.
    [Show full text]
  • In the Dirac Tradition
    In the Dirac tradition The late Richard Feynman - 'a way of looking at things so they appear not so mysterious'. It was Paul Dirac who cast quantum mechanics into the form we now use, and many generations of theo­ reticians openly acknowledge his in­ fluence on their thinking. When Dirac died in 1984, St. John's College, Cambridge, his base for most of his lifetime, institu­ ted an annual lecture in his memory at Cambridge. The first lecture, in 1986, attracted two heavyweights - Richard Feynman (see page 1) and Steven Weinberg. Far from using the lectures as a platform for their own work, in the Dirac tradition they pre­ sented stimulating material on deep underlying questions. (The lectures - 'Elementary Parti­ cles and the Laws of Physics' by Richard P. Feynman and Steven Weinberg - are published by Cam­ bridge University Press.) Richard Feynman When I was a young man, Dirac was my hero. He made a break­ through, a new method of doing tic quantum mechanics. However, cles, then to get the new wave- physics. He had the courage to the puzzle of negative energies that function from the old you must put simply guess at the form of an the equation presented, when it in a minus sign. It is easy to de­ equation, the equation we now call was solved, eventually showed monstrate that if Nature was non- the Dirac equation, and to try to in­ that the crucial idea necessary to relativistic, if things started out that terpret it afterwards. Maxwell in his wed quantum mechanics and relati­ way then it would be that way for day got his equations, but only in vity together was the existence of all time, and so the problem would an enormous mass of 'gear antiparticles.
    [Show full text]
  • Is the Quantum State Real in the Hilbert Space Formulation?
    Is the Quantum State Real in the Hilbert Space Formulation? Mani L. Bhaumik Department of Physics and Astronomy, University of California, Los Angeles, USA. E-mail: [email protected] Editors: Zvi Bern & Danko Georgiev Article history: Submitted on November 5, 2020; Accepted on December 18, 2020; Published on December 19, 2020. he persistent debate about the reality of a quan- 1 Introduction tum state has recently come under limelight be- Tcause of its importance to quantum informa- The debate about the reality of quantum states is as old tion and the quantum computing community. Almost as quantum physics itself. The objective reality underly- all of the deliberations are taking place using the ele- ing the manifestly bizarre behavior of quantum objects gant and powerful but abstract Hilbert space formal- is conspicuously at odds with our daily classical phys- ism of quantum mechanics developed with seminal ical reality. The scientific outlook of objective reality contributions from John von Neumann. Since it is commenced with the precepts of classical physics that rather difficult to get a direct perception of the events cemented our notion of reality for centuries. Discover- in an abstract vector space, it is hard to trace the ies starting in the last decade of the nineteenth century progress of a phenomenon. Among the multitude of revealing the uncanny quantum world in the microscopic recent attempts to show the reality of the quantum domain shook that perception. state in Hilbert space, the Pusey–Barrett–Rudolph With the particular exception of Einstein, who was the theory gets most recognition for their proof.
    [Show full text]
  • Strong Loophole-Free Test of Local Realism*
    Selected for a Viewpoint in Physics week ending PRL 115, 250402 (2015) PHYSICAL REVIEW LETTERS 18 DECEMBER 2015 Strong Loophole-Free Test of Local Realism* † Lynden K. Shalm,1, Evan Meyer-Scott,2 Bradley G. Christensen,3 Peter Bierhorst,1 Michael A. Wayne,3,4 Martin J. Stevens,1 Thomas Gerrits,1 Scott Glancy,1 Deny R. Hamel,5 Michael S. Allman,1 Kevin J. Coakley,1 Shellee D. Dyer,1 Carson Hodge,1 Adriana E. Lita,1 Varun B. Verma,1 Camilla Lambrocco,1 Edward Tortorici,1 Alan L. Migdall,4,6 Yanbao Zhang,2 Daniel R. Kumor,3 William H. Farr,7 Francesco Marsili,7 Matthew D. Shaw,7 Jeffrey A. Stern,7 Carlos Abellán,8 Waldimar Amaya,8 Valerio Pruneri,8,9 Thomas Jennewein,2,10 Morgan W. Mitchell,8,9 Paul G. Kwiat,3 ‡ Joshua C. Bienfang,4,6 Richard P. Mirin,1 Emanuel Knill,1 and Sae Woo Nam1, 1National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA 2Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1 3Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA 4National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA 5Département de Physique et d’Astronomie, Université de Moncton, Moncton, New Brunswick E1A 3E9, Canada 6Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, 100 Bureau Drive, Gaithersburg, Maryland 20899, USA 7Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA 8ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 9ICREA-Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain 10Quantum Information Science Program, Canadian Institute for Advanced Research, Toronto, Ontario, Canada (Received 10 November 2015; published 16 December 2015) We present a loophole-free violation of local realism using entangled photon pairs.
    [Show full text]
  • The Quantum World the Disturbing Theory at the Heart of Reality
    The Quantum World The disturbing theory at the heart of reality NEW SCIENTIST 629462_The_Quant_World_Book.indb 3 3/11/17 12:37 PM First published in Great Britain in 2017 by John Murray Learning. An Hachette UK company. Copyright © New Scientist 2017 The right of New Scientist to be identified as the Author of the Work has been asserted by it in accordance with the Copyright, Designs and Patents Act 1988. Database right Hodder & Stoughton (makers) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, or as expressly permitted by law, or under terms agreed with the appropriate reprographic rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, John Murray Learning, at the address below. You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer. British Library Cataloguing in Publication Data: a catalogue record for this title is available from the British Library. Library of Congress Catalog Card Number: on file. ISBN: 978 1 47362 946 2 eISBN: 978 1 47362 947 9 1 The publisher has used its best endeavours to ensure that any website addresses referred to in this book are correct and active at the time of going to press. However, the publisher and the author have no responsibility for the websites and can make no guarantee that a site will remain live or that the content will remain relevant, decent or appropriate.
    [Show full text]
  • Physics in the Second Half of the Twentieth Century JOSÉ MANUEL SÁNCHEZ RON
    the world after the revolution: physics in the second half of the twentieth century JOSÉ MANUEL SÁNCHEZ RON The great revolutions of the twentieth the state of movement of the observer, and mass, m, century is equivalent to energy, E (the famous expression E= m·c2, where c represents the speed of light), opened During the fi rst half of the twentieth century—actually, new doors for understanding the physical world. For the fi rst quarter—there were two major scientifi c example, this theory helped explain how it was possible revolutions. Those cognitive cataclysms took place in that radioactive elements (uranium, polonium, radium, physics, and are known as the relativist and quantum thorium) that had been studied for the fi rst time by revolutions. They are respectively related to the special Henri Becquerel (1852-1908) and Marie (1867-1934) and general theories of relativity (Einstein 1905a, and Pierre Curie (1859-1906), emit radiation in a 1915), and quantum mechanics (Heisenberg 1925, continuous manner with no apparent loss of mass. Schrödinger 1926). And then there was the general theory of relativity, which explained gravity by converting space—actually, Relativity four-dimensional space-time—into something curved, Much has been written, and will be written in the and with variable geometry! It was immediately future, about the importance of those theories apparent that, compared to Newton’s universal and their effect on physics as a whole, even before gravitation, Einstein’s new theory made it much easier to the middle of the century. Created to resolve the understand perceptible phenomena in the solar system increasingly evident “lack of understanding” between (it solved, for example, a century-old anomaly in the Newtonian mechanics and the electrodynamics of movement of Mercury’s perihelion).
    [Show full text]
  • User:Guy Vandegrift/Timeline of Quantum Mechanics (Abridged)
    User:Guy vandegrift/Timeline of quantum mechanics (abridged) • 1895 – Wilhelm Conrad Röntgen discovers X-rays in experiments with electron beams in plasma.[1] • 1896 – Antoine Henri Becquerel accidentally dis- covers radioactivity while investigating the work of Wilhelm Conrad Röntgen; he finds that uranium salts emit radiation that resembled Röntgen’s X- rays in their penetrating power, and accidentally dis- covers that the phosphorescent substance potassium uranyl sulfate exposes photographic plates.[1][3] • 1896 – Pieter Zeeman observes the Zeeman split- ting effect by passing the light emitted by hydrogen through a magnetic field. Wikiversity: • 1896–1897 Marie Curie investigates uranium salt First Journal of Science samples using a very sensitive electrometer device that was invented 15 years before by her husband and his brother Jacques Curie to measure electrical Under review. Condensed from Wikipedia’s Timeline of charge. She discovers that the emitted rays make the quantum mechanics at 13:07, 2 September 2015 (oldid surrounding air electrically conductive. [4] 679101670) • 1897 – Ivan Borgman demonstrates that X-rays and This abridged “timeline of quantum mechancis” shows radioactive materials induce thermoluminescence. some of the key steps in the development of quantum me- chanics, quantum field theories and quantum chemistry • 1899 to 1903 – Ernest Rutherford, who later became that occurred before the end of World War II [1][2] known as the “father of nuclear physics",[5] inves- tigates radioactivity and coins the terms alpha and beta rays in 1899 to describe the two distinct types 1 19th century of radiation emitted by thorium and uranium salts. [6] • 1859 – Kirchhoff introduces the concept of a blackbody and proves that its emission spectrum de- pends only on its temperature.[1] 2 20th century • 1860–1900 – Ludwig Eduard Boltzmann produces a 2.1 1900–1909 primitive diagram of a model of an iodine molecule that resembles the orbital diagram.
    [Show full text]
  • 0707.0401.Pdf
    J.S. Bell’s Concept of Local Causality Travis Norsen Marlboro College, Marlboro, VT 05344∗ (Dated: March 3, 2011) John Stewart Bell’s famous 1964 theorem is widely regarded as one of the most important devel- opments in the foundations of physics. It has even been described as “the most profound discovery of science.” Yet even as we approach the 50th anniversary of Bell’s discovery, its meaning and impli- cations remain controversial. Many textbooks and commentators report that Bell’s theorem refutes the possibility (suggested especially by Einstein, Podolsky, and Rosen in 1935) of supplementing or- dinary quantum theory with additional (“hidden”) variables that might restore determinism and/or some notion of an observer-independent reality. On this view, Bell’s theorem supports the orthodox Copenhagen interpretation. Bell’s own view of his theorem, however, was quite different. He in- stead took the theorem as establishing an “essential conflict” between the now well-tested empirical predictions of quantum theory and relativistic local causality. The goal of the present paper is, in general, to make Bell’s own views more widely known and, in particular, to explain in detail Bell’s little-known mathematical formulation of the concept of relativistic local causality on which his theorem rests. We thus collect and organize many of Bell’s crucial statements on these topics, which are scattered throughout his writings, into a self-contained, pedagogical discussion including elaborations of the concepts “beable”, “completeness”, and “causality” which figure in the formula- tion. We also show how local causality (as formulated by Bell) can be used to derive an empirically testable Bell-type inequality, and how it can be used to recapitulate the EPR argument.
    [Show full text]
  • The Age of Entanglement
    CONTENTS Title Page Dedication Epigraph List of Illustrations A Note to the Reader Introduction: Entanglement 1: The Socks 1978 and 1981 The Arguments 1909–1935 2: Quantized Light September 1909–June 1913 3: The Quantized Atom November 1913 4: The Unpicturable Quantum World Summer 1921 5: On the Streetcar Summer 1923 6: Light Waves and Matter Waves November 1923–December 1924 7: Pauli and Heisenberg at the Movies January 8, 1925 8: Heisenberg in Helgoland June 1925 9: Schrödinger in Arosa Christmas and New Year’s Day 1925–1926 10: What You Can Observe April 28 and Summer 1926 11: This Damned Quantum Jumping October 1926 12: Uncertainty Winter 1926–1927 13: Solvay 1927 14: The Spinning World 1927–1929 15: Solvay 1930 INTERLUDE: Things Fall Apart 1931–1933 16: The Quantum-Mechanical Description of Reality 1934–1935 The Search and the Indictment 1940–1952 17: Princeton April–June 10, 1949 18: Berkeley 1941–1945 19:Quantum Theory at Princeton 1946–1948 192 20: Princeton June 15–December 194 21: Quantum Theory 1951 22: Hidden Variables and Hiding Out 1951–1952 23: Brazil 1952 24: Letters from the World 1952 25: Standing Up to Oppenheimer 1952–1957 26: Letters from Einstein 1952–1954 Epilogue to the Story of Bohm 1954 The Discovery 1952–1979 27: Things Change 1952 28: What Is Proved by Impossibility Proofs 1963–1964 29: A Little Imagination 1969 30: Nothing Simple About Experimental Physics 1971–1975 31: In Which the Settings Are Changed 1975–1982 Entanglement Comes of Age 1981–2005 32: Schrödinger’s Centennial 1987 33: Counting to Three 1985–1988 34: “Against ‘Measurement’” 1989–1990 35: Are You Telling Me This Could Be Practical? 1989–1991 36: The Turn of the Millennium 1997–2002 37: A Mystery, Perhaps 1981–2006 Epilogue: Back in Vienna 2005 Glossary Longer Summaries Notes Bibliography Acknowledgments Permissions About the Author Copyright For my father If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
    [Show full text]